Numerical Analysis Seminars: The numerical simulation of wavepackets in a 2D nonlinear boundary layer
22 February 2008 14:15 in eScience 034
Results from two-dimensional direct numerical simulations of the governing equations that model incompressible fluid flow over a flat plate are presented. The Navier-Stokes equations are cast in a novel velocity-vorticity formulation (see Davies and Carpenter (2001)) and discretized with a mixed pseudospectral and compact finite-difference scheme in space, and a three-level backward-difference scheme in time.
A method to determine the envelope of a wavepacket (from numerical data) was developed. Based on the usual Hilbert Transform, new stages were incorporated to ensure a smooth envelope was found when the wavepacket was asymmetric.
The early transitional stages of the Blasius flow (flow over a flat plate with zero streamwise pressure gradient) are investigated with particular regard to a weakly nonlinear effect called wave-envelope steepening. Blasius flow is linearly unstable and so-called Tollmien-Schlichting modes develop. As nonlinearities become significant, the envelope of the wavepacket starts to develop differently at its leading and trailing edges. Numerical results presented here show that the envelope becomes steeper at the leading edge than it is at the trailing edge.
The effect of a non-zero streamwise pressure gradient on wave-envelope steepening is investigated by using Falkner-Skan profiles in place of the Blasius profile.
Natural transition is triggered by randomly-modulated waves. A disturbance with a randomly-modulated envelope was modelled and its effect on wave-envelope steepening was studied.
The higher-order Ginzburg-Landau equation was used to model the evolution of an envelope of a wavepacket disturbance. These results gave good qualitative comparison with the direct numerical simulations.
Finally, in preparation for developing a three-dimensional nonlinear version of the code, the discretization of one of the governing equations (the Poisson equation) was extended to three dimensions. Results from this new three-dimensional version of the Poisson solver show good agreement with those from an iterative solver, and also demonstrate the robustness of the numerical scheme.
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